PCA Eigenvector Signs and Scaling Eigenportfolio Weights
Summary
The document addresses two practical interpretation questions about principal component analysis in statistical arbitrage. First, it asks whether loadings in the leading eigenvector must all be positive and whether negative loadings can be changed to positive values individually. The accepted answer explains that an eigenvector’s overall sign is arbitrary: the entire vector may be multiplied by minus one, but taking absolute values of individual entries changes its direction and meaning. A second answer notes that sign ambiguity also applies when retaining multiple components, provided each component is reversed as a whole.
Second, the discussion considers why eigenportfolio weights derived from a correlation matrix divide eigenvector coefficients by each variable’s standard deviation. The answer distinguishes PCA on covariance from PCA on correlation: correlation PCA finds directions in standardized, scaled returns, so converting those directions into portfolio weights requires scaling. The exchange is conceptual and points readers toward further references; it does not provide a full derivation or implementation procedure.
Key ideas
- An eigenvector and its negation represent the same principal component direction.
- Changing individual loading signs or taking absolute values changes the vector and is not an equivalent sign convention.
- Each component can be reversed as a whole without changing its interpretation.
- Correlation-matrix PCA operates on standardized returns, so portfolio weights require adjustment for variable standard deviations.
- The exchange explains the distinction conceptually but does not give a full derivation or implementation guide.
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Full text
# Some clarifications on eigenvectors and eigenvalues from PCA # Some clarifications on eigenvectors and eigenvalues from PCA Could somebody tell me whether suggestions in bold true or not? Q # 1: http://www.math.nyu.edu/faculty/avellane/AvellanedaLeeStatArb071108.pdf ``` Chapter 2.2 Interpretation of the eigenvectors/eigenportfolios ``` This paper says that loadings in the maximal eigenvector need to be all positive and should not change sign, what if i have negative ones, can i force them to be positive always by simply taking them by module, e.g. MathAbs(Vector) ? Q # 2 : The same paper also defines weights for eigenportfolio in this way : ``` Q[i] = EigVecCoef[i] / StdDev[i] // Page # 10 paragraph 2.2 in the doc above ``` There is also another paper that says that eigenvector is an angle (or direction) of the portfolio's spread which allows to map current portfolio's spread to initial axes (dimensions) : http://georgemdallas.wordpress.com/2013/10/30/principal-component-analysis-4-dummies-eigenvectors-eigenvalues-and-dimension-reduction/ So i do not understand - why do i need to divide each value in eigenvector by standard deviation to calculate weights if this portfolio is already mapped to initial axes? ## Answer by Eric Brady (score 1, accepted) https://quant.stackexchange.com/a/15615 Q #1: I'm not sure if you have the answer quite right. The signs for the loadings are arbitrary, but you cannot take the absolute value. You can multiply by -1. Q #2: It might be helpful to think about what PCA is actually doing. This paper might be helpful: http://arxiv.org/pdf/1404.1100v1.pdf (A Tutorial on Principal Component Analysis by Jonathon Shlens). The key point is towards the end, where the author explains the relationship of SVD with PCA. "We can conclude that finding the principal components amounts to finding an orthonormal basis that spans the column space..." of the data matrix. But this quote is referring to finding the principal components using the covariance matrix. The authors of the paper you attached use the correlation matrix. This is essentially finding a set of variables that spans the space of scaled returns. Thus you need to scale the weights of your eigenvectors. You may also find this piece helpful - it seems to discuss a similar application with a little more derivation: Some clarifications on eigenvectors and eigenvalues from PCA ## Answer by Anonymous (score 0) https://quant.stackexchange.com/a/15609 A # 1 : Several replies from the following topic answer my Q # 1 - yes, if I take only one dimension after PCA then I can simply make all vectors positive, hence take only absolute values by module. If I take several dimensions then entire vector needs to be reverted and each value inside particular vector can be multiplied by -1 because reversion of orthogonal vectors does not change their meaning, the only requirement in this case is if one value in vector was changed then all other need to be changed too. Time series of PCA - Sign change in factor loadings
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